−
−
−
χ
χ
χ
χ
1
1
1
1
(8)
Maximum implosion
pressure of a single
bubble
Neglecting the thermal conduction which, however, should be very low due
to the short implosion time, for the occurring temperature
An evaluation of the equations as a function of the ratio partial pressure of the
gas inside the bubble p
G
to standard pressure p
N
, which is a measure for the
gas content of the bubble, supplies the following maximumvalues for T = 293 K
and χ = 1.4:
19
Part 3 ⋅ L351 EN
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O
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A
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0
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/
1
1
Maximum occurring
temperature
p
p
G
N
[-]
p
i max
N
mm
2

`
,
− χ
χ
1
1
(9)
Due to the pressure gradients in the ambient fluid or the influence of rigid boun-
daries, the cavitation bubbles generally deviate from the spherically symmetric
shape. They implode forming a microjet, as high-speed films show and as indi-
cated according to [11] in Fig. 8 for three typical cases. Plesset and Chapman
[12] have analyzed the implosion process and found the proportionality
for the jet velocity
Lauterborn [13] used a rotating mirror camera with a frame rate of
900,000 pictures per second to showthe formation of a microjet and determin-
ed maximumjet velocities between 50 and 100 m/s, which confirmed the valu-
es that have been theoretically determined by several authors.
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i
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v
p p
jet
V
F
≈
−
∞
ρ
(10)
The jet velocity of the
microjet reaches
up to 100 m/s
Asymmetric bubble
implosion leads to the
formation of a microjet
Fig. 8: Diagram illustrating the collapse of a bubble
Bubble moving into a
region of higher pressure
Bubble collapsing near the
wall of a rigid boundary
Hemispherical bubble
clinging to the wall of a
rigid boundary
The pressure created by a fluid jet impact on the wall of a rigid boundary can be
easily estimated if the fluid jet is assumed to be a elastic deformable solid.
The application of the principle of conservation momentum results in
For water (ρ
F
≈ 1000 kg/m
3
, c
F
≈ 1500 m/s), pressure surges with amplitudes
between 750 and 1500 N/mm² are reached with the jet velocities mentioned
above. Depending on the size of the bubble (radius R), the surge
lasts between several microseconds and several milliseconds. The effect of one
single surge is limited to an area of only a few micrometers in diameter.
The damage to the surface is caused to a considerable extent by the impact of
the striking fluid jet and by the shock wave of the imploding bubble. Most prob-
ably, the high temperature in the imploding bubble itself is also a key factor.
21
Part 3 ⋅ L351 EN
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t
R
c
surge
F
·
2
(12)
p c v
c
c
surge F F jet
F F
W W
· ⋅ ⋅ ⋅
⋅
⋅
+ ρ
ρ
ρ
1
and because w c c
w F F
⋅ >> ⋅ ρ
p c v
surge F F jet
≈ ⋅ ⋅ ρ (11)
In his publication [14], Lauterborn showed that the collapsing bubble,
which acquires a toroidal shape, becomes unstable and collapses at se-
veral locations.
These centers of collapse are the starting point for shock waves
which lead to the characteristic toroidal pattern of damage. The ex-
tent of damage depends mainly on the dimensionless parameter for
distance γ = s/R
max
, where s stands for the distance between the cen-
ter point of the bubble and the boundary at the maximum bubble radi-
us R
max
. A correlation between liquid jet and pattern of damage arises
only when γ < 0.7, if the bubble is already located at the solid bounda-
ry before it collapses. In this case, the jet reaches unobstructed the sur-
face and leaves an impression on the surface [14].
Knapp [15] showed for a stationary cavitation zone according to Fig. 9
that only one out of 30,000 bubbles implodes near the wall, thus ha-
ving a damaging effect. The number of high-energy surges per cm²
reaches its maximum at the end of the cavitation zone (Fig. 9).
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80
[Surges/cm
2
]
60
40
20
0
0 1 2 3 [–]
I
I
0
0 I
0
Fig. 9: Number of high-energy surges in a stationary cavitation zone
Cavitation intensity
While the effect of a collapsing bubble can be very well described by theoret-
ical models, the cavitation intensity of a cavitation zone can currently only be
described in terms of quality. The following parameters are significant:
4x
FZ
value of the control valve
4operating pressure ratio x
F
4pressure difference p
2
− p
V
4geometry of the control valve downstream of the restriction
4gas contents of the fluid
4fluid viscosity
4surface tension of the fluid
4fluid density
The difference between the pressure in the bubble and the pressure at the site
of implosion acts as the driving force when the bubble collapses, see equa-
tions (10) and (11). Since the pressure in the bubble is almost equal to the va-
por pressure, and the ambient pressure at the site of implosion corresponds
approximately to the downstream pressure, the damaging effect of the bub-
ble increases as the difference p
2
– p
V
increases. If the difference between
downstream pressure and vapor pressure is less than 2 bars, as often is the
case in heat supply systems, there is no significant material erosion when
non-corrosive media are used. The cavitation zone extends and the number
of cavitation bubbles grows as the difference between the pressure ratio x
F
and the valve-specific coefficient x
FZ
for incipient cavitation increase. On the
one hand, this leads to an increase in the damaging effect at first. On the ot-
her hand, as the difference between x
F
and x
FZ
increases, the duration of the
bubble growth phase increases as well. Consequently a larger amount of ga-
ses dissolved in the fluid diffuses into the bubble.
23
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The gas diffusion
increases as the differ-
ence between x
F
and
x
FZ
increases
The diffused gas is transported with the bubble into the cavitation zone and is
released when the bubble collapses. As a result, the compressibility K of the
fluid increases in the area of the collapse, while the density ρ
F
decreases. The
sound velocity drops as a consequence
The result according to equation (11) is a reduction in pressure surge ampli-
tudes causing the eroding effect of the collapsing bubbles to decrease. The
cavitation wear under otherwise identical conditions is smaller with supersa-
turated liquids than with undersaturated liquids due to the same reasons. Be-
sides, an extreme undersaturation causes a drop in the critical pressure at
which incipient cavitation occurs to values far belowthe vapor pressure. This
means the incipient cavitation first occurs when the pressure ratio is greater
than x
FZ
and is less intense due to the lack of cavitation nuclei. The bubbles
implode all the more energetically, the larger the pressure gradient downst-
reamof the restriction is. At the given operating conditions, the pressure gra-
dient is determined by the geometry of the control valve. The gradient is
particularly large when the free jet hits the valve body wall close the restricti-
on. The energy dissipation is low as turbulent mixing has just started in this
case, and the pressures close to the wall can reach values similar to the up-
streampressure p
1
. The driving force for the bubble implosion is then appro-
ximately proportional (p
1
– p
V
) and not as described above (p
2
– p
V
).
A reduction in viscosity causes, when all other conditions remain the same, an
increase in the number and size of the bubbles. Additionally, the kinematic im-
pulse of the microjet at low liquid viscosity is greater than at high viscosity. If
laminar flow conditions are assumed due to the small size for the microjet, then
the jet velocity is inversely proportional to the viscosity.
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c
K
F
F
·
⋅
1
ρ
(13)
Diffused gas increases
the compressibility of
the fluid
The eroding effect
decreases as the com-
pressibiliy increases
Turbulent mixing causes
the pressure to drop
Then the much quoted exponential correlation between the material erosion ∆m
and jet velocity v follows
for the distinct influence of operating viscosity on the material erosion
The pressures occurring when the bubble implodes are proportional accord-
ing to equation (7) to the root of the density. This is why the erosion rate is
particularly high when cavitation occurs in mercury or liquefied metals.
The surface tension or capillarity makes the pressure in the bubble increase
according to equation (1). Therefore, liquids with smaller surface tension
than water cavitate at pressure ratios lower than x
FZ
. When the conditions
are otherwise the same, the size and number of cavitation bubbles increases
as the surface tension lessens, while the driving force during bubble implo-
sion is reduced. Technical literature and other sources supplies deviating de-
tails about the effect of surface tension on material erosion. It may, however,
be assumed that the surface tension influences incipient cavitation consider-
ably, but its effect on material erosion is small when the cavitation is pro-
nounced.
A certain judgement, that experienced engineers are most likely to have, is
needed to assess cavitation erosion as the effects of the above described pa-
rameters affect and overlap each other.
25
Part 3 ⋅ L351 EN
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O
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A
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1
1
∆
∆
m
m
v
v
k
k
1
2
1
2
1 8 ·
|
.

`
,
· ... (14)
∆
∆
m
m
k
1
2
2
1
·
|
.

`
,

ν
ν
(15)
The erosion rate becomes
more rapid as medium
density increases
Cavitation erosion
The material surface, depending on its structure, is deformed, loosened and
eventually eroded in particles in various ways due to the frequent strain from
the pressure waves created by the microjet occurring when the bubble col-
lapses. Fig. 10 shows how cavitation erosion is generally subdivided into
three areas.
4In area I, termed the incubation period, a loss in weight is not yet measura-
ble.
4Area II is characterized by an almost constant erosion rate. Areas and
depths of the material erosion increase with time.
4In area III, the surface that is already strongly fractured reduces the proba-
bility for an implosion close to the surface as it acts as a kind of protective
cushion; the material erodes at a much slower rate.
26
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I
II
III
Fig. 10: The course of development of cavitation erosion
L
o
s
s
i
n
w
e
i
g
h
t
Time
Cavitation erosion is
subdivided into three
areas
During the incubation period, the surface of ductile metals first only under-
goes elastic deformation and then plastic deformation. Dents and bulges
arise. Their number increases as time passes. After the deformability of the
material is exceeded at the end of the incubation period, fractures occur and
individual particles break off.
The deformation phase does not take place to a great extent with brittle met-
als due to the high density of dislocation obstacles. As a consequence, inter-
nal tensions form that exceed the material strength at the end of the
incubation phase. Fractures and ruptures occur that cause a monotonously
increasing weight loss. In cast iron, the graphite phase is eroded when the
cavitation strain starts, meaning an incubation period can only be mentioned
in conjunction with pearlite or ferrite erosion.
After the graphite nodules have been eliminated from spheroidal cast iron,
the soft ferrite phase flows into the emptied troughs. The pearlite acts as a
supporting frame and slows down surface deformation. The end of the incu-
bation period is reached when the ferrite breaks off at the trough edges after
strong deformation and the pearlite material areas break away due to mate-
rial fatigue.
In cast iron with lamellar graphite, the ferrite is eroded after the incubation
time elapses without much plastic deformation because after the graphite has
been removed, the contact between the ferrite blocks remains restricted to a
few metal bridges which cannot withstand the constant strain.
In ceramic material, microfractures start to appear right from the beginning
of the strain without undergoing any noticeable plastic deformation before-
hand. As the strain continues, the density of fractures continuously grows. Af-
ter the incubation time has elapsed, the fractures have spread and joined
each other and breakages occur. The incubation in plastics is similar to duc-
tile metals where plastic deformation, formation of cracks and spread of
cracks occurs. In crystalline plastics, these structure faults are formed by dis-
location, in amorphous plastics by the breaking up of atomic bonds.
27
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In all materials, surface roughness in the area where the bubbles implode
leads to a noticeable increase in material erosion since the machining
notches help the bubbles to implode energetically and the ridges of the
roughly machined surfaces resist the imploding bubbles less than even sur-
faces would. The influence of the surface roughness must though be observed
in conjunction with the microhardness achieved by machining. According to
[16], an austenitic steel (EN material no. 1.4919, corresponding to
AISI 316 H) with a milled surface has a microhardness HV
0.05
of 430,
whereas it has just 246 with an electrolytically polished surface.
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Cavitation corrosion
Apart from mechanical strain, corrosive and stress corrosive influences de-
termine the erosion rate of the material. The combination of cavitation erosi-
on and corrosion where the aggressive components can intensify each other
is termed cavitation corrosion. The effect of impacting liquid jets intensifies
the corrosive attack as the forming top and passivation layers are immedi-
ately worn away, causing the high initial corrosion rate typical of bright me-
tal surfaces to be kept as long as the strain lasts.
A beneficial factor for the corrosion process is also the free oxygen which the
cavitation bubbles absorb by diffusion, also from undersaturated liquids,
while they are growing and which they then release in the cavitation area
when they implode.
Additionally, the ions of a corrosive medium are likely to interact with the
crystallographic slip steps and fractures of areas that have undergone plastic
deformation due to liquid pressure waves. This leads to an accelerated mate-
rial destruction due to intensified fracture formation or due to fracture
spreading.
29
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Cavitation resistance
Magnetostrictive transducers (Fig. 12) as well as cavitation chambers
(Fig. 11), electromagnetic, piezoelectric and ultrasonic vibrating instruments
are mostly used in laboratories to investigate cavitation resistance of materi-
als. In the magnetostrictive transducer, nickel laminations are excited to vi-
brate by high-frequency alternating currents.
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Fig. 11: Cavitation chamber
Fig. 12: Magnetostrictive transducer
Heating/cooling
Field coil
Sample
Sonotrode
Sample
These vibrations are transferred to a coupled sonotrode, which is designed to
vibrate in response at the predetermined frequency. The material samples
are attached either to the free end of the rod and/or face it at a defined dis-
tance.
The liquid surrounding the free end of the rod cannot follow the high-fre-
quency rod oscillations due to its mass inertia, causing liquid cavities in the
form of small bubbles to be formed which erode the material when they im-
plode. The test set ups allow cavitation resistance of various materials to be
investigated under controlled conditions, making it possible to specify a cor-
relation between the mechanical material characteristics and the erosion
rate.
According to the investigations of R. Garcia and F.G. Hammitt [17], the cavi-
tation resistance K
R
is proportional to the deformation energy (introduced by
Hobbs) up to the point of fracture UR (ultimative resilience – Fig. 13).
31
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σ
B
σ
ε
B
ε
UR=
.
σ
B
.
ε
B
ε
2
=
1
.
σ
B
2
2
1
σ
S
[N/mm
2
]
[mm]
Fig. 13: Material property UR
K UR
R
≈ (16)
Cavitation resistance K
R
UR is a combined material property that describes to some extent the energy
which could be stored elastically in a material per unit of volume if the yield
point could be raised to the level of tensile stress.
Berger [16] describes the cavitation resistance with the relation
In the equation, E stands for elasticity module, HV for Vickers hardness, R
m
for tensile strength and R
p 0.2
for the top yield point or the strength at 0.2%
elasticity. The good match of the values calculated according to the equa-
tion (17) with values determined in a cavitation chamber (nozzle sample)
can be seen in Fig. 14.
32
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K
R W HV
E R
R
m
p
≈
⋅ ⋅
⋅
1071 0 125 1971
0 562
0 2
0 618
. . .
.
.
.
(17)
14
K
RCK 45
12
measured
calculated
material
10
8
6
G
G
G

4
0
G
G
G

4
0

(
a
u
s
t
e
n
i
t
i
c
)
C
1
0
S
t

5
0
-
2
C
k

4
5
C
4
5

t
e
m
p
e
r
e
d
9
0

M
n

V
8
X

2
2

C
r

N
i

1
7
X

9
0

C
r

M
o

V

1
8
X

5

C
r

N
i

1
8

9
3
4

C
r

N
i

M
o

6
1
6

M
n

C
r

5
4
2

M
n

C
r

4
4
2

M
n

C
r

4

t
e
m
p
e
r
e
d
S
p
e
c
i
a
l

b
r
a
s
s
C
u

A
I

1
0

N
i
A
I

C
u

M
g

1
A
I

Z
n

M
g
4
2
0
K
R
Fig. 14: Cavitation resistance
The equations (16) and (17) are only suitable for assessing metals with suffi-
cient elasticity which allow the yield point and deformation energy to be de-
termined in a tensile test.
The applicability of the equations (16) and (17), that were derived from tests
with clear water and hydraulic fluid, is still restricted to cases using less cor-
rosive fluids. If cavitation corrosion as described in the section on cavitation
corrosion is expected, it is recommendable upon selecting the material to
pay attention primarily to the good corrosion resistance and then to take the
cavitation resistance into consideration.
33
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Avoiding cavitation
As already explained on page 14, cavitation is avoided when the pressure
ratio x
F
at the control valve is smaller than the corresponding x
FZ
value for all
operating cases. If the operating pressure ratio x
F
is kept small by a clever
plant layout, the selection of the control valve decides whether the ratio
x
F
< x
FZ
is fulfilled and a cavitation-free flow can be guaranteed.
The x
FZ
value, which can theoretically be 1 (no pressure recovery) is deter-
mined considerably by the Carnot impact loss which is a function of the ope-
ning ratios (nominal size cross-section to restriction cross-section). The
impact loss and the x
FZ
value with it, increases as the opening ratio increases,
whereas the K
v
value drops as the opening ratio increases, which can also be
expressed by K
v
Control valve/K
vs
Ball valve.
The x
FZ
value range of various types of control valves is shown in Fig. 15 as a
function of the K
v
value of the valve related to the K
vs
value of a completely
open ball valve in the same nominal size.
The bandwidth of the x
FZ
value range is expressed to a great extent by the hy-
draulic diameter
the shape of the valve plug and seat and the number of pressure reduction
stages.
For the free cross-section A = DN
2
⋅ π/4 of a non-reduced ball valve, the cir-
cumference U = DN · π and thus the hydraulic diameter is equal to the nomi-
nal size DN.
Another expression for the hydraulic diameter is the valve style modifier F
d
used in the IEC 60534-2-1 standard [25].
34
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The x
FZ
value increases
as the opening ratio
increases
d
A
U
h
· ⋅ 4 (18)
In a linear valve with a parabolic plug, the free cross-section is ring-shaped
at small loads and the hydraulic radius is just a fraction of the nominal diam-
eter DN. This results in a large surface area of the free jet downstream of the
restriction which leads to an intensive impulse exchange with the surround-
ing medium and causes high pressure losses.
As can be seen in Fig. 15, ball valves only allow the mediumto be controlled
without cavitation at small pressure ratios. Butterfly valves and rotary plug
valves are slightly better, whereas linear valves allow control without any
cavitation even at high pressure ratios when the plug is designed accord-
ingly.
Control valves which can be fitted with anti-cavitation trims (Fig. 16) for re-
ducing cavitation and multi-stage axial plugs (Fig. 17) should be given a
special mention here.
35
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x
FZ [-]
[-]
K
v Control valve
K
vs Ball valve
0
0 1
d
h
< DN
d
h
= DN
Linear valves
Rotary valve/ Butterfly valve
Ball Valves
Fig. 15: Control valve operating range without cavitation
Linear valves allow
control without any
cavitation even at
high pressure ratios
The system illustrated in Fig. 16 is a seat and plug trim specially develo-
ped for cases where cavitation occurs and designed to be fitted in existing
valves. The trimhas been optimized in a series of flow simulations and in-
tensive tests. The plug is double guided in the body to prevent mechanical
vibrations, and the plug contours have been designed for better flow cha-
racteristics. The seat diameter has not been reduced to keep the hydraulic
diameter as small as possible. In combination with the special shape of
the plug and seat, this is particularly effective. Additionally, a maximum
of four attenuation plates can be integrated into the seat to additionally
increase the x
FZ
value at high valve loads [18].
36
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Fig. 16: AC trim system (SAMSON)
Fig. 17: Multi-stage axial plug (SAMSON Type 3255 Valve)
For an initial rough assessment, it is possible to assume that the x
FZi
values of
each individual stage are the same in a valve with a multi-stage plug. Then,
the x
FZ
value is obtained for an n-stage control valve according to
and for the K
v
value ratio of each stage
The K
v
value of an n-stage control valve arises from the K
vi
values of each
individual stage according to
The number of stages n required to manage an operating pressure ratio x
F
free of cavitation is obtained when the x
FZi
value is known from
37
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( )
x x
FZ FZi
n
· − − 1 1 (19)
K
K
x
vi
vi
FZi
+
· −
1
1 (20)
K
K K K
v
v v vn
·
+ + +
1
1 1 1
1
2
2
2 2
...
(21)
n
x
x
F
FZi
·
−
−
lg( )
lg( )
1
1
(22)
The relationships are easier to understand when looking at Fig. 18. It can be
seen that a five-stage valve whose individual stages each have an x
FZi
value
of 0.3 reaches an x
FZ
value of > 0.8, its K
v
value is however just 30% of the
K
vn
value of the last stage. The K
vn
value corresponds approximately to the
K
vs
value possible in a one-stage valve of the same nominal size. Besides the
axial stage plug, the radial stage plugs have proven themselves well in prac-
tice (Fig. 19, left).
The x
FZ
value of the perforated plug (Fig. 19, center) is determined by the
opening ratio and the hydraulic radius of the largest hole. The division of
holes should be kept to at least three hole diameters to avoid the free jets from
joining together before the impulse exchange with the surrounding medium
has finished. Just relatively small K
vs
values (or large opening ratios) are pos-
sible especially with equal percentage characteristics. But they lead to high
x
FZ
values as shown in Fig. 15. Valves with perforated cages (Fig. 19, right)
function similarly to valves with perforated plugs.
38
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i
n
0.1
0.1
0.2
0.4
0.6
0.8
1.0
K
v
/K
vn
0.2 0.3 0.4
n=5
n=4
n=3
n=2
n=1
x
FZi
= 0.1
0.5 0.6 0.7 0.8 0.9 1.0 x
FZ
0.2 0.3 0.4 0.5 0.6
Fig. 18: x
FZ
values of multi-stage valves
A special design (deviations, restriction cascades) of the ducts allows the
countermeasures which increase the x
FZ
value, for example, high opening
ratio, small hydraulic radius and multi-stage throttling, to be combined in
one valve. High x
FZ
values (i.e. low pressure recovery), on the one hand, re-
quire large opening ratios, and thus relatively small K
v
values. Large K
v
val-
ues related to the nominal size, on the other hand, always lead to relatively
small x
FZ
values. The x
FZ
values are listed in the data sheets of control valve
manufacturers. Specifications that are outside of the value range shown in
Fig. 15 should be viewed critically though.
39
Part 3 ⋅ L351 EN
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Fig. 19: Plug designs
Operation with cavitation
Since x
FZ
values necessary for cavitation-free operation cannot always be
achieved, or the control valves are too susceptible to dirt or too expensi-
ve, making their use uneconomical, valves are often operated in condi-
tions where cavitation occurs. A troublefree operation with a satisfactory
service life can be guaranteed though if the effects of cavitation are taken
into consideration on sizing and selecting the valve. To ensure that the
valve body walls are outside the area at risk of erosion, a larger valve
should be selected than required by the valve sizing simply based on the
seat diameter. To achieve this, the nominal diameter should be selected to
ensure that a certain outlet velocity is not exceeded. It can then be assu-
med that the cavitation zones will not reach up to the valve walls, and that
the bubbles implode without any harm and with little effect (apart from
noise).
The following table helps to show whether erosion damage due to cavita-
tion is to be expected. This is the case when x
F
> x
Fcrit,cav
as well as when
p
1
-p
2
> ∆p
crit,cav
[18]. Additionally, a nominal valve size should be selec-
ted to ensure that the outlet velocity does not exceed 4 m/s.
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Valve design
x
Fcrit,cav
[-]
∆p
crit,cav
[bar]
Single-stage linear valves
0,7 15
Single-stage linear valves with
stellited or hardened trim
0,7 25
3-stage linear valves 1,0 100
5-stage linear valves 1,0 200
Rotary plug valves 0,4 10
Butterfly and ball valves 0,25 5
Table 2: Limit values for preventing cavitation erosion
The sealing slope and plug surface of parabolic plugs are particularly at
risk from erosion (Fig. 20).The area around the sealing edge should be
given a Stellite facing. A hard facing of the whole surface is recommen-
ded when the cavitation intensity increases.
V-port plugs are exposed less to cavitation attacks than parabolic plugs
under the same operating conditions. Due to the diverted jet (Fig. 21), the
cavitation zones do not stick to the plug surface, meaning the cavitation
bubbles implode almost without any effect.
41
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Fig. 20: Areas of a parabolic plug at risk from erosion
Fig. 21: Diverted jet occurring with a V-port plug
The diverted jet pre-
vents the V-port plug
from being eroded
The sealing slope and
plug surface of parabo-
lic plugs are particulary
at risk from erosion
The valve seat is not exposed to the cavitation attack when the medium
flows against the closing direction. However, if the medium flows in the
reverse direction to protect the valve body walls and plug facing (risk of
pressure surge waves), the seat and plug surface are then particularly at
risk from erosion. A satisfactory service life can only be achieved in this
case by using highly resistant materials.
The perforated plug (Fig. 22) is better suited for mediumflow in the reverse di-
rection. A steep pressure gradient that makes the cavitation bubbles implode
forms in the center of the plug, i.e. in sufficient distance away fromthe surface
due to the collision of the partial flows. However, the cavitation zone moves out
of the plug when the differential pressure rises, causing the body floor to
erode. The above mentioned pressure ratios and differential pressures should
only be used as a reference for identifying the cavitation intensity. The limits
can be considerably reduced as a result of cavitation corrosion, especially
when corrosive fluids are used.
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A rising differential
pressure can cause the
body floor to erode
A medium flowing in
the reverse direction
increases the risk of
erosion
Fig. 22: Cavitation zone with a perforated plug
Influence on the hydraulic
characteristics
The continuity of the fluid phase is interrupted when cavitation occurs and the
dynamic interaction between flow and its restriction is affected. Additionally,
cavitation makes the compressibility of the fluid increase locally and in this
way reduces the sound velocity (equation 13). The density of the fluid, too, is
drastically reduced by the bubble volume in the area of the restriction as the
pressure ratio rises.
These effects limit the flow rate in control valves (choked flow) if a certain dif-
ferential pressure (∆p
max
) is exceeded as shown in Fig. 23.
43
Part 3 ⋅ L351 EN
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Cavitation changes the
density and compressi-
bility of a medium
x
FZ
K
C
F
L
∆p
Q
∆p
max
Fig. 23: The valve-specific factor F
L
The critical pressure p
DK
is reached at the vena contracta at the differential
pressure ∆p
max
, which is below the vapor pressure according to measure-
ments by Stiles [19] (Fig. 24).
Stiles found the relationship shown in Fig. 25 for the ratio between critical
pressure at the vena contracta and the vapor pressure
from measurements using Frigen 12, which can also be approached using
the following equation:
44
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p
1
p
2
p
V
p
DK
∆p
DK
∆p
max
Fig. 24: The critical pressure p
DK
The critical pressure is
below vapor pressure
F
p
p
F
DK
V
· (23)
F
p
p
F
V
C
· − 0 96 0 28 . . (24)
Baumann [20] introduced the valve-specific factor F
L
for the ratio bet-
ween the smallest differential pressure across the valve at which the cho-
ked flow starts, ∆p
max
and the critical differential pressure at the vena
contracta, ∆p
DK.
F
L
value is the pressure recovery factor. It is determined froma flowrate measu-
rement as in Fig. 23. In this measurement, the upstream pressure is kept cons-
tant and the downstreampressure is reduced until choked flowstarts. Adetailed
description of the procedure is specified in IEC 60534, Part 3.
45
Part 3 ⋅ L351 EN
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F
p
p
L
DK
2
·
∆
∆
max
(25)
0.60 0 0.40 0.20
0.6
0.7
0.8
0.9
1.0
1.00 0.80
F
F
p
V
p
C
Fig. 25: F
F
factor used to determine the critical pressure ratio
F
L
: the factor for
pressure recovery
The maximum value for differential pressure at which the flow rate is
achieved due to cavitation is obtained with the F
L
value and equation (23).
The first deviation in flow rate from the general measurement equation
is indicated by the pressure ratio.
The K
c
value is often referred to as the incipient cavitation index. Yet, inci-
pient cavitation already occurs, as shown in Fig. 23, at the much smaller
pressure ratio x
FZ
. If both coefficients are confused with another, it causes se-
rious mistakes in the assessment of the pressure recovery, the cavitation in-
tensity and noise emission.
The distribution of pressure on the surface of the closure member is also
changed by the change in flow pattern due to cavitation.
46
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( )
∆p F p F p
L F V max
· − ⋅
2
1
(26)
K
p
p p
C
V
·
−
∆
1
(28)
Q K p
v
· ⋅ ⋅ ∆
ρ
ρ
0
(27)
Therefore, the hydraulic torque in butterfly valves, which is calculated with
cavitation-free flow according to
and works in the closing direction, struggles against a limit value as the pres-
sure ratio grows. In equation (29), D is the disc diameter, C
T
the torque coef-
ficient dependent on the opening angle ϕ, and H the static differential
pressure p
1
– p
2
across the butterfly valve, increased by the impact pressure
of the flow velocity v:
In Fig. 26, the torque for an opening angle ϕ of 60° is shown as a function of
the pressure ratio x
F
at constant upstream pressure.
47
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Hydraulic torque with
cavitation-free flow
600
100
0 0.5
F
L
²
1.0
x
F

M
T [Nm]
[-]
Fig. 26: Course of torque in a butterfly valve
H p
F
· +
ρ
ν
2
2
(30)
M C D H
T T
· ⋅ ⋅
3
(29)
The torque reaches its maximumat approximately the same differential pres-
sure at which also the choked flow occurs (equation 26). As the pressure ra-
tio (or ∆p) further increases, the torque drops slightly at first and then reaches
a limit value irrespective of the differential pressure.
When the limit value is reached, the cavitation zone stretches over the whole
downstream pressure side, causing the pressure distribution on the inlet side
to be approximately constant with the vapor pressure p
V
irrelevant of how
the pressure ratio increases.
The flow forces around the plug change due to cavitation even in linear
valves. However, this change in force is insignificant, except in self-operated
regulators where the plug forces must be carefully balanced out by means of
pressure balancing.
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Torque reaches its
maximum at
choked flow
Changes in fluid properties
Not just the surface of the hydraulic components, but even the fluid itself, is
exposed to extreme loads when gas-filled cavitation bubbles implode.
For example, when bubbles implode in hydraulic fluids, temperatures can
occur that are sufficient under certain circumstances to ignite bubbles con-
taining air and oil vapor. This process, which is called the microdiesel effect,
leads to an accelerated ageing of hydraulic fluids. Cavitation increases the
free gas content in the fluid: parts of the gas dissolved in the fluid diffuse into
the cavitation bubbles during the growth phase.
The gas parts are released when the bubbles implode and locally increase
the compressibility of the fluid. This is accompanied by a reduction in sound
velocity, meaning pressure surge and sound propagation calculations in
pipelines where cavitation occurs is made more difficult. Fig. 27 shows the
sound velocity in water as a function of the bubble concentration according
to [21].
49
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The microdiesel effect
leads to an accelerated
aging of hydraulic
fluids
10
4
[m/s]
[%]
10
3
10
2
10
1
0.0001 0.1 100
1080m/s
1500m/s
p

=

1
0

b
a
r p

=

1

b
a
r
p

=

0
.
1

b
a
r
340m/s
108m/s
c
F
Fig. 27: Sound velocity as a function of the bubble concentration
If oxygen is also released by the cavitation process, the oxidizing effect of the
fluid increases. The contamination of the medium caused by cavitation-in-
duced material erosion should not be underestimated in closed circuits.
50
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Cavitation noise
The pressure peaks induced by the bubble implosions cause material erosion
as well as a loud typical noise. The theoretical approaches to explain how
noise develops are based on individual bubbles which implode concentri-
cally in an infinitely expanded fluid and without affecting each other. The
cavitation bubble can then be regarded as an isotropic radiator of zero or-
der (monopole source) which creates in the far field the sound pressure
In the equation, V(t) is the rate of change in bubble volume which can be de-
termined approximately using the Rayleigh-Plesset differential equation, and
r is the distance between the sound source and the point of observation. The
spectral distribution of the energy E(f) of the noise created by a bubble is ob-
tained fromthe squared Fourier transformof the pressure over time in the far
field
while neglecting the retarded time t
r
c
f
−
|
.

⋅ ⋅ ⋅
−
−∞
+∞
∫
ρ π
π
2
4 2
2
(33)
The spectral energy rises at first with the frequency f raised to the power of
four and reaches a maximum at a frequency whose the inverse ratio is ap-
proximately the same as the collapsing time of the cavitation bubble. As a
consequence of the collapsing time being directly proportional to the bubble
radius, the maximum noise is shifted to lower frequencies as the bubble ra-
dius increases. At frequencies above the maximum, the spectral energy de-
creases at f
–2/5
.
If the bubble collapses are observed as random events not connected with
one another, whose frequency follows a Poisson distribution, it is possible to
derive the total spectral sound energy E
total
(f) fromthe spectral energy of indi-
vidual events E(f) and a mean number of bubble collapses per unit of time n.
These theoretical considerations made for individual bubbles apply approxi-
mately when cavitation just starts. As the cavitation progresses, distinctive
cavitation zones form in which the effects of each individual bubble overlap
each other, and the contribution of each bubble to the entire noise depends
on the history of the neighboring bubble.
More extensive investigations therefore observe the cavitating fluid as a
quasi-continuum of density
In the equation, C is the total volume of all bubbles per volume unit. Lyamshev
[22] discovered on this basis that the sound intensity of a cavitation zone
equals the flow velocity raised to the power of four, or is proportional to the
square of the differential pressure.
52
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E f n E f total ( ) ( ) · ⋅ (34)
Observations of individ-
ual bubbles only apply
to incipient cavitation
ρ
ρ
·
+
F
C ( ) 1
(35)
The theoretical approaches were extended and refined by various authors.
Despite this, no one has yet successfully described the exceptionally complex
correlations in the cavitation zone downstream of a control valve for various
media and pressure ratios.
Therefore, the currently valid equations as per the German directive
VDMA 24422 to predict noise emission of cavitating flows of control valves
are not based on hydrodynamic and thermodynamic models, but instead de-
scribe the course of internal sound power level on the basis of measurements
related to x
F
, z, ρ
F
and ∆p.
In this case, neither the influence of surface tension nor viscosity nor the influ-
ence of the gas contents are taken into account. The influence of the density
and the differential pressure important for the course of the bubble implosion
are also only reproduced inexactly in the empirical VDMAequations which are
based mainly on measurements with cold water (p
V
≈ 0 bar, ρ
F
≈1000 kg/m
3
).
The VDMA calculation methods [23] introduced in 1979 and improved in
1989 are, however, well proven and recognized worldwide.
Note: The IEC 60534-8-4 international standard is currently being revised.
The main features of the new standard are shown in [25].
53
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The equations accor-
ding to VDMA are ba-
sed on empirical data
p p
x
x
p
V
F
F
2
1
− ·
−
⋅ ∆ (36)
When cavitation (x
F
>> x
FZ
) occurs, the acoustic power emitted in the octave
band range of 0.5 to 8 kHz in the pipeline is obtained according to
The first row of the equation (37) represents the sound power level which is
caused by the turbulent flow noise. The second row serves to calculate the
sound power component which arises when the flow noise caused by the tur-
bulence overlaps the cavitation noise resulting from the temporal static bub-
ble collapse. Fig. 28 shows the course of the standardized sound power for
standard valves with parabolic plugs (∆L
F
= 0) in relation to the operating
pressure ratio with x
FZ
as a parameter. When x
F
> x
FZ
applies, the noise
emission rises steeply, reaches a maximum and drops back to the sound po-
wer level caused by the turbulence when x
F
= 1.
The drop in sound power level at high pressure ratios is based, on the one
hand, on the compressibility of the fluid that increases with x
F
(see section on
avoiding cavitation) and on the reduction of the driving force p
2
– p
V
when
the bubble implodes (equation 10). Cavitation behavior deviating from the
calculation can be taken into consideration by the valve manufacturer by
specifying a valve-specific correction coefficient ∆L
F
= f (x
F
, y).
The spectral distribution of the internal sound power level depends on the de-
sign, pressure ratios, load and x
FZ
value of the valve. According to
VDMA 24422, the spectral distribution can be reproduced for practical
application in the octave band range 0.5 to 8 kHz irrelevant of the operating
condition using a noise spectrum that drops 3 dB per octave.
54
Control Valves ⋅ Cavitation in Control Valves
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L
W p
F
Wi
F
F L
· + ⋅
⋅ ⋅
⋅
134 4 10
2
. lg
∆ η
ρ
( )
Lwi
x x
x
x
x
L
FZ F
F
x
F
FZ
F
FZ
− ⋅
⋅ −
⋅
−
−
+ 120
1 1001
1
0 0625
0 8
.
.
lg
.
∆
with p F p F p
L F V
≤ − ⋅
2
1
( ) (37)
Several measurements performed by SAMSON resulted, however, in a sound
power spectrumas shown in Fig. 29. At incipient cavitation (x
F
= x
FZ
), the sound
power level radiated in the individual octave bands is approximately the same;
the low frequencies dominate as the pressure ratio rises.
55
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η
F
.
W
.
∆
p
ρ
F
.
F
L
2
L
W
i
–
1
0

hydrodynamics.
SAMSON AG ⋅ 03/11
7
. the British physicist Lord Rayleigh was asked to investigate what caused fast-rotating ship propellers to erode so quickly. however. In 1917. for non-invasive operations in the field of medicine and for breaking down agglomerates in the textile finishing industry. optics. regarded to be a destructive phenomenon for the most part. break up pollutants and dissolve out minerals from organic material. For example. a laser beam creates a plasma in liquids which causes the liquid to evaporate creating a cavity. cavitation is used to break down molecules and bacteria cell walls. plasma physics and chemistry. cavitation effects can be applied usefully for cleaning surfaces. Furthermore. This effect is. was the source of the problem. In sewage treatment plants. In addition to pump rotors. thermodynamics. Despite numerous investigations into the subject of cavitation in the years that followed. He discovered that the effect of cavitation. already proved in experiments by Reynolds in 1894.Part 3 ⋅ L351 EN
Introduction
Cavitation has been a familiar phenomenon for a long time particularly in shipping. This is no wonder considering the complexity of the process involving the areas of acoustics. cavitation can arise in hydrodynamic flows when the pressure drops. In this way. many of the accompanying effects have still not yet been completely explained. control valves are particularly exposed to this problem since the static pressure at the vena contracta even at moderate operating conditions can reach levels sufficient for cavitation to start occurring in liquids. Cavitation can be caused in a fluid by energy input. Ultrasonic waves can be used to induce complex high-frequency alternating compression and rarefaction phases in liquids which cause cavitation.

Control Valves ⋅ Cavitation in Control Valves

The consequences for a control valve as well as for the entire control process vary and are often destructive:

Cavitation
Cavitation shall be generally understood as the dynamic process of the formation and implosion of cavities in fluids.Cavitation occurs, for instance, when high flow velocities cause the local hydrostatic pressure to drop to a critical value which roughly corresponds to the vapor pressure of the fluid. This causes small bubbles filled with steam and gases to form. These bubbles finally collapse when they reach the high-pressure areas as they are carried along by the liquid flow. In the final phase of bubble implosion, high pressure peaks are generated inside the bubbles and in their immediate surroundings. These pressure peaks lead to mechanical vibrations, noise and material erosion of surfaces in walled areas. If cavitation is severe, the hydraulic valve coefficients as well as the fluid properties change.

SAMSON AG ⋅ 03/11

9

Control Valves ⋅ Cavitation in Control Valves

Incipient cavitation
The term ‘cavitation’ is derived from the Latin verb cavitare meaning ‘to hollow out’, thus referring to the formation of cavities. To create cavities in fluids, the fluid must first be expanded and then ruptured. Theoretically, fluids can absorb high tensile strengths or negative pressures. Ackeret [1] estimates the negative pressure required to cause cavitation in pure water at 20 °C, based on the minimum of the van der Waals curve, and receives a theoretically possible tensile strength of 104 bar which corresponds approximately to the reciprocal compressibility of water.
The tensile strength of the medium is reduced by the disturbance

Inhomogeneities (disturbances) in the quasi-crystalline structure of water, however, reduce the possible tensile strengths by minimum one order of magnitude. In all probability, submicroscopic accumulations of steam or gas molecules are created at these disturbances with the molecules being in an unstable equilibrium with the fluid. In the case of external tensile strengths (negative pressure), these nuclei can exceed a critical diameter and then grow spontaneously as steam is formed.

p 1500 1200

[bar]

Tensile strength

900 600 300 0 T
SAMSON AG ⋅ V74/ Dob/Zin

0 20 40 60 80 100

[ C]

o

Fig. 1: Theoretical tensile strength values for perfect water

10

2. As shown in Fig. If these values were actually reached for industrial fluids. i. 2 also shows one of the many anomalies of water. resulting in the theoretical tensile strength values for perfect water shown in Fig. The highest values known until now were derived by measuring centrifugal force as a function of the temperature with extremely pure water. Becker and Döring [2] determined the probability for a critical nucleus to occur in dependence of the temperature. cavitation in connection with hydraulic systems would not be a matter for discussion. the highest value to be achieved was only at 280 bars [3]. 1. Fig. 2: Tensile strength values derived from measurements for pure water
SAMSON AG ⋅ 03/11
11
. the strong reduction of tensile strength near the freezing point which is caused by the formation of water crystals.Part 3 ⋅ L351 EN
By means of statistical examination.
p 280 240 200 160 120 80 40 0 0
[bar]
T 200 400 [oC]
Fig.e.

Control Valves ⋅ Cavitation in Control Valves
The discrepancy between the theoretically and experimentally (under ideal conditions) determined strength values shows that the microscopic bubbles
Cavitation nuclei are decisive for the occurrence of cavitation
filled with gas and steam (cavitation nuclei). which exist in the fluid and whose existence in water can be explained according to the model by Harvey [4].e. i.
12
SAMSON AG ⋅ V74/ Dob/Zin
G 2α − R R3
(3)
. the following can be derived from the general gas law for spherical nuclei:
pG =
N ⋅ RG ⋅ T G = 3 4 R ⋅ π ⋅R3 3
(2)
and with (1)
p − pV =
with G being proportional to the gas volume contained in the nucleus. are a decisive factor in the occurrence of cavitation. Spherical cavitation nuclei are stable when the fluid pressure p acting on the bubble surface and the partial pressure 2α/ R resulting from the surface tension are in equilibrium with the sum of the partial pressures inside the bubble. If you consider the volume change of the bubble to be isothermal. R is the radius of the bubble and α is the capillary constant. the vapor pressure pV and the pressure of the enclosed gas volume pG:
pG + pV =
2α +p R
(1)
In this equation.

Instead. and so forth. spectrum of nuclei. Generally. which in turn build up a pressure field that causes even smaller bubbles to collapse. Lehmann and Young [5] examined the phenomenon of cavitation hysteresis in depth and found that the end of
Incipient cavitation starts just below vapor pressure
cavitation can be more easily reproduced than incipient cavitation. the cavitation
SAMSON AG ⋅ V74/ Dob/Zin
coefficient xFZ [6] has proven useful. it is impossible to determine a liquid’s spectrum of nuclei in advance for most applications. The pressure distribution along the body contour is derived from Bernoulli’s equation so that a relationship between the minimum pressure pmin and the critical pressure can be stated according to equation (4). As a result. i. In particular cases. the critical pressure at which cavitation stops is higher than the critical pressure at incipient cavitation. Their sudden growth accelerates the ambient fluid and reduces the pressure locally. This pressure reduction causes the next smaller bubbles to collapse.
14
.
Stalling flows occur in control valves
In case of stalling flows as they occur in control valves.
The cavitation coefficient xFZ In the case of less viscous liquid flows around streamlined bodies. at which static pressure cavitation begins. The velocity distribution of these types of flow can be calculated on the basis of the potential theory if the flow conditions are known. Therefore. start to cavitate first. this is just below the vapor pressure. especially the temperature. it is common practice in control valve sizing to describe the critical state of the cavitation nuclei at incipient cavitation by means of the vapor pressure of the liquid. it therefore depends on the state of the liquid. the potential theory cannot be used to determine the minimum pressure.Control Valves ⋅ Cavitation in Control Valves
Large nuclei. the content of dissolved gases and the surface tension. In practice.e. bubbles with a large G value. the internal friction compared to the pressure may be frequently neglected.

4):
x FZ =
p1 − p 2 p 1 − p min
(5)
Since the minimum pressure occurs in one of the unsteady vortex cores downstream of the restriction.Part 3 ⋅ L351 EN
p1
p2
p1
p2 pmin pV
Fig.
SAMSON AG ⋅ 03/11
Minimum pressure occurs downstream of the restriction
15
. the ratio of the external pressure difference (p1 − p2) to the internal pressure difference (p1 − pmin) for all cavitation-free operating states equals a valve-specific value xFZ (Fig. it cannot be determined by direct measurement. in a control valve. 4: Distribution of pressure in the valve
It is based on the assumption that.

However. these relationships can strictly speaking only be applied to media which conform to the test medium water regarding their nuclei spectrum.Part 3 ⋅ L351 EN
If a valve’s xFZ values are known over the entire travel range. The experimental xFZ values should therefore be rounded to full five hundredths to account for the accuracy limits of the process. since the difference pV – pcrit according to equation (4) is not covered by the operating pressure ratio xF.3 [–]
SAMSON AG ⋅ 03/11
Fig. (Fig. In case of an operating pressure ratio xF < xFZ. it can be determined in advance for all operating pressure ratios
xF =
∆p p1 − pV
(6)
whether cavitation effects are to be expected.
The operating case xF = xFZ indicates incipient cavitation
28
ml l
Content of dissolved gases
24 20 16 12 8 4 0 0.1 xF 0.2 0. 7: Relation between the pressure ratio xF and the gas content
17
. surface tension and viscosity. 7). Oldenziel [8] clearly showed this by measuring the pressure ratio xF at incipient cavitation as a function of the gas content of water (Fig. there is no danger of cavitation occurring. 6). when xF ≥ xFZ. a stationary cavitation zone builds up whose expansion is roughly proportional to the difference (xF – xFZ).

8: Diagram illustrating the collapse of a bubble
20
.
Bubble moving into a region of higher pressure
Bubble collapsing near the wall of a rigid boundary
Hemispherical bubble clinging to the wall of a rigid boundary
SAMSON AG ⋅ V74/ Dob/Zin
Fig. Plesset and Chapman [12] have analyzed the implosion process and found the proportionality
v jet ≈
p ∞ − pV ρF
(10)
for the jet velocity Lauterborn [13] used a rotating mirror camera with a frame rate of
The jet velocity of the microjet reaches up to 100 m/s
900.Control Valves ⋅ Cavitation in Control Valves
Asymmetric bubble implosion leads to the formation of a microjet
Due to the pressure gradients in the ambient fluid or the influence of rigid boundaries. which confirmed the values that have been theoretically determined by several authors.000 pictures per second to show the formation of a microjet and determined maximum jet velocities between 50 and 100 m/s. 8 for three typical cases. the cavitation bubbles generally deviate from the spherically symmetric shape. They implode forming a microjet. as high-speed films show and as indicated according to [11] in Fig.

Part 3 ⋅ L351 EN
The pressure created by a fluid jet impact on the wall of a rigid boundary can be easily estimated if the fluid jet is assumed to be a elastic deformable solid.
SAMSON AG ⋅ 03/11
21
. cF ≈ 1500 m/s). pressure surges with amplitudes between 750 and 1500 N/mm² are reached with the jet velocities mentioned above. the high temperature in the imploding bubble itself is also a key factor. The damage to the surface is caused to a considerable extent by the impact of the striking fluid jet and by the shock wave of the imploding bubble. The effect of one single surge is limited to an area of only a few micrometers in diameter. Most probably. Depending on the size of the bubble (radius R). the surge
t surge =
2R cF
(12)
lasts between several microseconds and several milliseconds. The application of the principle of conservation momentum results in
p surge = ρF ⋅ c F ⋅ v jet ⋅
ρF ⋅ c F +1 ρW ⋅ c W
and because w ⋅ c w >> ρ F ⋅ c F
p surge ≈ ρF ⋅ c F ⋅ v jet
(11)
For water (ρF ≈ 1000 kg/m3.

These centers of collapse are the starting point for shock waves which lead to the characteristic toroidal pattern of damage. thus having a damaging effect.7. if the bubble is already located at the solid boundary before it collapses. In this case. The extent of damage depends mainly on the dimensionless parameter for distance γ = s/Rmax . which acquires a toroidal shape. Lauterborn showed that the collapsing bubble. 9).000 bubbles implodes near the wall.Control Valves ⋅ Cavitation in Control Valves
In his publication [14]. becomes unstable and collapses at several locations. where s stands for the distance between the center point of the bubble and the boundary at the maximum bubble radius Rmax. The number of high-energy surges per cm² reaches its maximum at the end of the cavitation zone (Fig. 9: Number of high-energy surges in a stationary cavitation zone
22
SAMSON AG ⋅ V74/ Dob/Zin
.
Knapp [15] showed for a stationary cavitation zone according to Fig. the jet reaches unobstructed the surface and leaves an impression on the surface [14].
[Surges/cm2] 80
60 0 I0
40
20 I I0 0 1 2 3 [–]
0
Fig. A correlation between liquid jet and pattern of damage arises only when γ < 0. 9 that only one out of 30.

On the other hand. the damaging effect of the bubble increases as the difference p2 – pV increases. Consequently a larger amount of gases dissolved in the fluid diffuses into the bubble.
23
. If the difference between downstream pressure and vapor pressure is less than 2 bars. this leads to an increase in the damaging effect at first. the cavitation intensity of a cavitation zone can currently only be described in terms of quality. and the ambient pressure at the site of implosion corresponds approximately to the downstream pressure. as often is the case in heat supply systems. see equations (10) and (11). The cavitation zone extends and the number of cavitation bubbles grows as the difference between the pressure ratio xF and the valve-specific coefficient xFZ for incipient cavitation increase. On the one hand. Since the pressure in the bubble is almost equal to the vapor pressure.Part 3 ⋅ L351 EN
Cavitation intensity
While the effect of a collapsing bubble can be very well described by theoretical models. there is no significant material erosion when non-corrosive media are used. the duration of the
SAMSON AG ⋅ 03/11
The gas diffusion increases as the difference between xF and xFZ increases
bubble growth phase increases as well. The following parameters are significant:
4 xFZ value of the control valve 4 operating pressure ratio xF 4 pressure difference p2 − pV 4 geometry of the control valve downstream of the restriction 4 gas contents of the fluid 4 fluid viscosity 4 surface tension of the fluid 4 fluid density
The difference between the pressure in the bubble and the pressure at the site of implosion acts as the driving force when the bubble collapses. as the difference between xF and xFZ increases.

The sound velocity drops as a consequence
cF =
1 K ⋅ ρF
(13)
The eroding effect decreases as the compressibiliy increases
The result according to equation (11) is a reduction in pressure surge amplitudes causing the eroding effect of the collapsing bubbles to decrease. the compressibility K of the fluid increases in the area of the collapse. Besides. The energy dissipation is low as turbulent mixing has just started in this case. the larger the pressure gradient downstream of the restriction is. Additionally. The cavitation wear under otherwise identical conditions is smaller with supersaturated liquids than with undersaturated liquids due to the same reasons.
24
. At the given operating conditions. an increase in the number and size of the bubbles. The gradient is particularly large when the free jet hits the valve body wall close the restricti-
Turbulent mixing causes the pressure to drop
on. then
SAMSON AG ⋅ V74/ Dob/Zin
the jet velocity is inversely proportional to the viscosity. when all other conditions remain the same.Control Valves ⋅ Cavitation in Control Valves
Diffused gas increases the compressibility of the fluid
The diffused gas is transported with the bubble into the cavitation zone and is released when the bubble collapses. the pressure gradient is determined by the geometry of the control valve. The driving force for the bubble implosion is then approximately proportional (p1 – pV) and not as described above (p2 – pV). A reduction in viscosity causes. As a result. and the pressures close to the wall can reach values similar to the upstream pressure p1. an extreme undersaturation causes a drop in the critical pressure at which incipient cavitation occurs to values far below the vapor pressure. The bubbles implode all the more energetically. the kinematic impulse of the microjet at low liquid viscosity is greater than at high viscosity. If laminar flow conditions are assumed due to the small size for the microjet. This means the incipient cavitation first occurs when the pressure ratio is greater than xFZ and is less intense due to the lack of cavitation nuclei. while the density ρF decreases.

. When the conditions are otherwise the same. 8 (14)
for the distinct influence of operating viscosity on the material erosion
∆m1  ν 2  =  ∆m 2  ν1 
k
(15)
The pressures occurring when the bubble implodes are proportional according to equation (7) to the root of the density. liquids with smaller surface tension than water cavitate at pressure ratios lower than xFZ. Technical literature and other sources supplies deviating details about the effect of surface tension on material erosion. This is why the erosion rate is particularly high when cavitation occurs in mercury or liquefied metals. while the driving force during bubble implosion is reduced. that experienced engineers are most likely to have.
SAMSON AG ⋅ 03/11
The erosion rate becomes more rapid as medium density increases
25
. is needed to assess cavitation erosion as the effects of the above described parameters affect and overlap each other. but its effect on material erosion is small when the cavitation is pronounced. A certain judgement.Part 3 ⋅ L351 EN
Then the much quoted exponential correlation between the material erosion ∆m and jet velocity v follows
∆m1  v 1  =  ∆m 2  v 2 
k
k = 1 . It may.. The surface tension or capillarity makes the pressure in the bubble increase according to equation (1). Therefore. be assumed that the surface tension influences incipient cavitation considerably. however. the size and number of cavitation bubbles increases as the surface tension lessens.

Fig.
4 In area III.
III
Loss in weight
II
I
Time
26
SAMSON AG ⋅ V74/ Dob/Zin
Fig.
4 Area II is characterized by an almost constant erosion rate.
4 In area I. the surface that is already strongly fractured reduces the probability for an implosion close to the surface as it acts as a kind of protective cushion. loosened and
Cavitation erosion is subdivided into three areas
eventually eroded in particles in various ways due to the frequent strain from the pressure waves created by the microjet occurring when the bubble collapses. a loss in weight is not yet measurable. termed the incubation period. the material erodes at a much slower rate. Areas and
depths of the material erosion increase with time. 10 shows how cavitation erosion is generally subdivided into three areas.Control Valves ⋅ Cavitation in Control Valves
Cavitation erosion
The material surface. depending on its structure. is deformed. 10: The course of development of cavitation erosion
.

The deformation phase does not take place to a great extent with brittle metals due to the high density of dislocation obstacles.Part 3 ⋅ L351 EN
During the incubation period.
27
. the graphite phase is eroded when the cavitation strain starts. After the incubation time has elapsed. fractures occur and individual particles break off. In cast iron with lamellar graphite. in amorphous plastics by the breaking up of atomic bonds. In crystalline plastics. microfractures start to appear right from the beginning of the strain without undergoing any noticeable plastic deformation beforehand. As a consequence. meaning an incubation period can only be mentioned in conjunction with pearlite or ferrite erosion. After the graphite nodules have been eliminated from spheroidal cast iron. the fractures have spread and joined each other and breakages occur. Their number increases as time passes. Fractures and ruptures occur that cause a monotonously increasing weight loss. The pearlite acts as a supporting frame and slows down surface deformation. In cast iron. In ceramic material. formation of cracks and spread of cracks occurs. As the strain continues. the density of fractures continuously grows. these structure faults are formed by disSAMSON AG ⋅ 03/11
location. the contact between the ferrite blocks remains restricted to a few metal bridges which cannot withstand the constant strain. The end of the incubation period is reached when the ferrite breaks off at the trough edges after strong deformation and the pearlite material areas break away due to material fatigue. The incubation in plastics is similar to ductile metals where plastic deformation. internal tensions form that exceed the material strength at the end of the incubation phase. After the deformability of the material is exceeded at the end of the incubation period. Dents and bulges arise. the soft ferrite phase flows into the emptied troughs. the surface of ductile metals first only undergoes elastic deformation and then plastic deformation. the ferrite is eroded after the incubation time elapses without much plastic deformation because after the graphite has been removed.

surface roughness in the area where the bubbles implode leads to a noticeable increase in material erosion since the machining notches help the bubbles to implode energetically and the ridges of the roughly machined surfaces resist the imploding bubbles less than even surfaces would.Control Valves ⋅ Cavitation in Control Valves
In all materials. The influence of the surface roughness must though be observed in conjunction with the microhardness achieved by machining.05 of 430. According to [16]. whereas it has just 246 with an electrolytically polished surface. 1.4919.
28
SAMSON AG ⋅ V74/ Dob/Zin
. an austenitic steel (EN material no. corresponding to AISI 316 H) with a milled surface has a microhardness HV0.

A beneficial factor for the corrosion process is also the free oxygen which the cavitation bubbles absorb by diffusion. This leads to an accelerated material destruction due to intensified fracture formation or due to fracture spreading.Part 3 ⋅ L351 EN
Cavitation corrosion
Apart from mechanical strain. while they are growing and which they then release in the cavitation area when they implode. also from undersaturated liquids.
SAMSON AG ⋅ 03/11
29
. Additionally. causing the high initial corrosion rate typical of bright metal surfaces to be kept as long as the strain lasts. corrosive and stress corrosive influences determine the erosion rate of the material. the ions of a corrosive medium are likely to interact with the crystallographic slip steps and fractures of areas that have undergone plastic deformation due to liquid pressure waves. The combination of cavitation erosion and corrosion where the aggressive components can intensify each other is termed cavitation corrosion. The effect of impacting liquid jets intensifies the corrosive attack as the forming top and passivation layers are immediately worn away.

making it possible to specify a correlation between the mechanical material characteristics and the erosion rate. 13).G. which is designed to vibrate in response at the predetermined frequency. B 2 ε
[mm] εB
SAMSON AG ⋅ 03/11
ε
Fig. . Hammitt [17]. According to the investigations of R. 13: Material property UR
31
. causing liquid cavities in the form of small bubbles to be formed which erode the material when they implode. Garcia and F. The material samples are attached either to the free end of the rod and/or face it at a defined distance.Part 3 ⋅ L351 EN
These vibrations are transferred to a coupled sonotrode.
Cavitation resistance KR
K R ≈ UR
(16)
σ σB σS
[N/mm2]
UR=
1. the cavitation resistance KR is proportional to the deformation energy (introduced by Hobbs) up to the point of fracture UR (ultimative resilience – Fig. The test set ups allow cavitation resistance of various materials to be investigated under controlled conditions. σ B εB 2 1 σ 2 = . The liquid surrounding the free end of the rod cannot follow the high-frequency rod oscillations due to its mass inertia.

If cavitation corrosion as described in the section on cavitation corrosion is expected.
SAMSON AG ⋅ 03/11
33
. it is recommendable upon selecting the material to pay attention primarily to the good corrosion resistance and then to take the cavitation resistance into consideration. is still restricted to cases using less corrosive fluids. The applicability of the equations (16) and (17).Part 3 ⋅ L351 EN
The equations (16) and (17) are only suitable for assessing metals with sufficient elasticity which allow the yield point and deformation energy to be determined in a tensile test. that were derived from tests with clear water and hydraulic fluid.

SAMSON AG ⋅ V74/ Dob/Zin
For the free cross-section A = DN2⋅ π/4 of a non-reduced ball valve. the circumference U = DN · π and thus the hydraulic diameter is equal to the nominal size DN. cavitation is avoided when the pressure ratio xF at the control valve is smaller than the corresponding xFZ value for all operating cases. The xFZ value range of various types of control valves is shown in Fig. If the operating pressure ratio xF is kept small by a clever plant layout. The bandwidth of the xFZ value range is expressed to a great extent by the hydraulic diameter
dh = 4 ⋅
A U
(18)
the shape of the valve plug and seat and the number of pressure reduction stages. 15 as a function of the Kv value of the valve related to the Kvs value of a completely open ball valve in the same nominal size.
34
. the selection of the control valve decides whether the ratio xF < xFZ is fulfilled and a cavitation-free flow can be guaranteed.Control Valves ⋅ Cavitation in Control Valves
Avoiding cavitation
As already explained on page 14. increases as the opening ratio increases. The
The xFZ value increases as the opening ratio increases
impact loss and the xFZ value with it. Another expression for the hydraulic diameter is the valve style modifier Fd used in the IEC 60534-2-1 standard [25]. which can also be expressed by Kv Control valve/Kvs Ball valve. whereas the Kv value drops as the opening ratio increases. which can theoretically be 1 (no pressure recovery) is determined considerably by the Carnot impact loss which is a function of the opening ratios (nominal size cross-section to restriction cross-section). The xFZ value.

whereas linear valves allow control without any cavitation even at high pressure ratios when the plug is designed accordingly. 15: Control valve operating range without cavitation
In a linear valve with a parabolic plug. 16) for reducing cavitation and multi-stage axial plugs (Fig. As can be seen in Fig.
Linear valves allow control without any cavitation even at high pressure ratios
SAMSON AG ⋅ 03/11
35
. the free cross-section is ring-shaped at small loads and the hydraulic radius is just a fraction of the nominal diameter DN. Control valves which can be fitted with anti-cavitation trims (Fig. This results in a large surface area of the free jet downstream of the restriction which leads to an intensive impulse exchange with the surrounding medium and causes high pressure losses. ball valves only allow the medium to be controlled without cavitation at small pressure ratios. 15. 17) should be given a special mention here. Butterfly valves and rotary plug valves are slightly better.Part 3 ⋅ L351 EN
xFZ 1
[-] Linear valves Rotary valve/ Butterfly valve Ball Valves
dh < DN dh = DN Kv Control valve Kvs Ball valve 0 0 1 [-]
Fig.

16 is a seat and plug trim specially developed for cases where cavitation occurs and designed to be fitted in existing valves.
Fig. In combination with the special shape of the plug and seat. The trim has been optimized in a series of flow simulations and intensive tests. The seat diameter has not been reduced to keep the hydraulic diameter as small as possible. and the plug contours have been designed for better flow characteristics. this is particularly effective. 16: AC trim system (SAMSON)
The system illustrated in Fig. Additionally.Control Valves ⋅ Cavitation in Control Valves
Fig. 17: Multi-stage axial plug (SAMSON Type 3255 Valve)
36
SAMSON AG ⋅ V74/ Dob/Zin
. The plug is double guided in the body to prevent mechanical vibrations. a maximum of four attenuation plates can be integrated into the seat to additionally increase the xFZ value at high valve loads [18].

+ 2 2 K v1 K v 2 K vn
(21)
The number of stages n required to manage an operating pressure ratio xF free of cavitation is obtained when the xFZi value is known from
n=
SAMSON AG ⋅ 03/11
lg(1 − x F ) lg(1 − x FZi )
(22)
37
. the xFZ value is obtained for an n-stage control valve according to
x FZ = 1 − (1 − x FZi )
n
(19)
and for the Kv value ratio of each stage
K vi = 1 − x FZi K vi +1
(20)
The Kv value of an n-stage control valve arises from the Kvi values of each individual stage according to
Kv =
1 1 1 1 + 2 +. it is possible to assume that the xFZi values of each individual stage are the same in a valve with a multi-stage plug. Then...Part 3 ⋅ L351 EN
For an initial rough assessment.

1 n=1
n=2
0. It can be seen that a five-stage valve whose individual stages each have an xFZi value of 0.2 0.4 0. Just relatively small Kvs values (or large opening ratios) are possible especially with equal percentage characteristics.5
0. its Kv value is however just 30% of the Kvn value of the last stage. The Kvn value corresponds approximately to the Kvs value possible in a one-stage valve of the same nominal size. But they lead to high xFZ values as shown in Fig.2
0.
Kv/Kvn 1.2
0.0 0.1 0. the radial stage plugs have proven themselves well in practice (Fig. Valves with perforated cages (Fig.5 0.1 0.3
0.7 0. center) is determined by the opening ratio and the hydraulic radius of the largest hole.4
0. Besides the axial stage plug.9 1.6 0.6 0.8 0. 19.8.4
xFZi = 0.Control Valves ⋅ Cavitation in Control Valves
The relationships are easier to understand when looking at Fig.0 xFZ
Fig. The xFZ value of the perforated plug (Fig. 18: xFZ values of multi-stage valves
38
SAMSON AG ⋅ V74/ Dob/Zin
. left).3 0. 15. The division of holes should be kept to at least three hole diameters to avoid the free jets from joining together before the impulse exchange with the surrounding medium has finished.3 reaches an xFZ value of > 0.6
n=3 n=4 n=5
0. right) function similarly to valves with perforated plugs. 19. 18.8 0. 19.

Part 3 ⋅ L351 EN
A special design (deviations. on the other hand. and thus relatively small Kv values. High xFZ values (i. to be combined in one valve. Large Kv values related to the nominal size. small hydraulic radius and multi-stage throttling. always lead to relatively small xFZ values. restriction cascades) of the ducts allows the countermeasures which increase the xFZ value.e. The xFZ values are listed in the data sheets of control valve manufacturers. 15 should be viewed critically though. high opening ratio. low pressure recovery). Specifications that are outside of the value range shown in Fig. for example.
Fig. 19: Plug designs
SAMSON AG ⋅ 03/11
39
. on the one hand. require large opening ratios.

0 1. To achieve this.0 0. valves are often operated in conditions where cavitation occurs. Additionally.cav [18].7 1. A troublefree operation with a satisfactory service life can be guaranteed though if the effects of cavitation are taken into consideration on sizing and selecting the valve. To ensure that the valve body walls are outside the area at risk of erosion.cav as well as when p1-p2 > ∆pcrit.cav [bar] 15 25 100 200 10
SAMSON AG ⋅ V74/ Dob/Zin
5
Table 2: Limit values for preventing cavitation erosion
40
.7 0. It can then be assumed that the cavitation zones will not reach up to the valve walls. This is the case when xF > xFcrit.4 0. and that the bubbles implode without any harm and with little effect (apart from noise). a larger valve should be selected than required by the valve sizing simply based on the seat diameter. a nominal valve size should be selected to ensure that the outlet velocity does not exceed 4 m/s.
Valve design Single-stage linear valves Single-stage linear valves with stellited or hardened trim 3-stage linear valves 5-stage linear valves Rotary plug valves Butterfly and ball valves
xFcrit.Control Valves ⋅ Cavitation in Control Valves
Operation with cavitation
Since xFZ values necessary for cavitation-free operation cannot always be achieved. the nominal diameter should be selected to ensure that a certain outlet velocity is not exceeded. or the control valves are too susceptible to dirt or too expensive. The following table helps to show whether erosion damage due to cavitation is to be expected. making their use uneconomical.25
∆pcrit.cav [-] 0.

21: Diverted jet occurring with a V-port plug
41
.The area around the sealing edge should be given a Stellite facing.Part 3 ⋅ L351 EN
Fig.
The sealing slope and plug surface of parabolic plugs are particulary at risk from erosion The diverted jet prevents the V-port plug from being eroded
SAMSON AG ⋅ 03/11
Fig. V-port plugs are exposed less to cavitation attacks than parabolic plugs under the same operating conditions. the cavitation zones do not stick to the plug surface. A hard facing of the whole surface is recommended when the cavitation intensity increases. 20). 21). meaning the cavitation bubbles implode almost without any effect. Due to the diverted jet (Fig. 20: Areas of a parabolic plug at risk from erosion
The sealing slope and plug surface of parabolic plugs are particularly at risk from erosion (Fig.

Fig. the cavitation zone moves out
A rising differential pressure can cause the body floor to erode
of the plug when the differential pressure rises. The above mentioned pressure ratios and differential pressures should only be used as a reference for identifying the cavitation intensity. A satisfactory service life can only be achieved in this case by using highly resistant materials. causing the body floor to erode. However. 22: Cavitation zone with a perforated plug
42
SAMSON AG ⋅ V74/ Dob/Zin
.e. especially when corrosive fluids are used. However. if the medium flows in the reverse direction to protect the valve body walls and plug facing (risk of pressure surge waves).Control Valves ⋅ Cavitation in Control Valves
The valve seat is not exposed to the cavitation attack when the medium
A medium flowing in the reverse direction increases the risk of erosion
flows against the closing direction. A steep pressure gradient that makes the cavitation bubbles implode forms in the center of the plug. The perforated plug (Fig. The limits can be considerably reduced as a result of cavitation corrosion. 22) is better suited for medium flow in the reverse direction. in sufficient distance away from the surface due to the collision of the partial flows. i. the seat and plug surface are then particularly at risk from erosion.

These effects limit the flow rate in control valves (choked flow) if a certain differential pressure (∆pmax) is exceeded as shown in Fig. is drastically reduced by the bubble volume in the area of the restriction as the pressure ratio rises.Part 3 ⋅ L351 EN
Influence on the hydraulic characteristics
The continuity of the fluid phase is interrupted when cavitation occurs and the dynamic interaction between flow and its restriction is affected. 23. too. 23: The valve-specific factor FL
SAMSON AG ⋅ 03/11
43
. The density of the fluid.
Cavitation changes the density and compressibility of a medium
Q
FL
KC xFZ ∆pmax
∆p
Fig. Additionally. cavitation makes the compressibility of the fluid increase locally and in this way reduces the sound velocity (equation 13).

28
pV pC
(24)
. 24). which is below the vapor pressure according to measurements by Stiles [19] (Fig. 25 for the ratio between critical pressure at the vena contracta and the vapor pressure
FF =
p DK pV
(23)
from measurements using Frigen 12. which can also be approached using the following equation:
44
SAMSON AG ⋅ V74/ Dob/Zin
FF = 0. Stiles found the relationship shown in Fig. 24: The critical pressure pDK The critical pressure is below vapor pressure
The critical pressure pDK is reached at the vena contracta at the differential pressure ∆pmax.Control Valves ⋅ Cavitation in Control Valves
p1 ∆pDK p2 pV pDK ∆pmax
Fig.96 − 0.

00
Fig.0 0. Part 3.40 0.80 1. the upstream pressure is kept constant and the downstream pressure is reduced until choked flow starts.
FF 1.8 0. In this measurement. It is determined from a flow rate measurement as in Fig. 23.9 0. A detailed description of the procedure is specified in IEC 60534.6 pV pC 0 0.20 0. ∆pmax and the critical differential pressure at the vena contracta.7 0.Part 3 ⋅ L351 EN
Baumann [20] introduced the valve-specific factor FL for the ratio between the smallest differential pressure across the valve at which the choked flow starts. ∆pDK.
FL: the factor for pressure recovery
FL 2 =
∆p max ∆p DK
(25)
FL value is the pressure recovery factor.60 0. 25: FF factor used to determine the critical pressure ratio
SAMSON AG ⋅ 03/11
45
.

∆p max = FL 2 ( p 1 − FF ⋅ p V )
(26)
The first deviation in flow rate from the general measurement equation
Q = K v ⋅ ∆p ⋅
ρ ρ0
(27)
is indicated by the pressure ratio. the cavitation intensity and noise emission. If both coefficients are confused with another. at the much smaller pressure ratio xFZ. incipient cavitation already occurs. 23. The distribution of pressure on the surface of the closure member is also changed by the change in flow pattern due to cavitation.
KC =
∆p p1 − pV
(28)
The Kc value is often referred to as the incipient cavitation index.Control Valves ⋅ Cavitation in Control Valves
The maximum value for differential pressure at which the flow rate is achieved due to cavitation is obtained with the FL value and equation (23).
46
SAMSON AG ⋅ V74/ Dob/Zin
. as shown in Fig. it causes serious mistakes in the assessment of the pressure recovery. Yet.

As the pressure ratio (or ∆p) further increases.Control Valves ⋅ Cavitation in Control Valves
Torque reaches its maximum at choked flow
The torque reaches its maximum at approximately the same differential pressure at which also the choked flow occurs (equation 26). When the limit value is reached. However.
48
SAMSON AG ⋅ V74/ Dob/Zin
. except in self-operated regulators where the plug forces must be carefully balanced out by means of pressure balancing. the cavitation zone stretches over the whole downstream pressure side. this change in force is insignificant. the torque drops slightly at first and then reaches a limit value irrespective of the differential pressure. causing the pressure distribution on the inlet side to be approximately constant with the vapor pressure pV irrelevant of how the pressure ratio increases. The flow forces around the plug change due to cavitation even in linear valves.

which is called the microdiesel effect. but even the fluid itself. 27: Sound velocity as a function of the bubble concentration
49
. meaning pressure surge and sound propagation calculations in pipelines where cavitation occurs is made more difficult. This is accompanied by a reduction in sound velocity. The gas parts are released when the bubbles implode and locally increase the compressibility of the fluid. Cavitation increases the free gas content in the fluid: parts of the gas dissolved in the fluid diffuse into the cavitation bubbles during the growth phase. temperatures can occur that are sufficient under certain circumstances to ignite bubbles containing air and oil vapor.1
100 [%]
Fig. p =
101 0.0001
SAMSON AG ⋅ 03/11
0.Part 3 ⋅ L351 EN
Changes in fluid properties
Not just the surface of the hydraulic components. is exposed to extreme loads when gas-filled cavitation bubbles implode. Fig. For example. leads to an accelerated ageing of hydraulic fluids. This process. when bubbles implode in hydraulic fluids.
The microdiesel effect leads to an accelerated aging of hydraulic fluids
cF 104
[m/s]
1500m/s
10
3
1080m/s
p = 10 ba r
108m/s 340m/s
p =
102
1 ba r 1 ba r 0. 27 shows the sound velocity in water as a function of the bubble concentration according to [21].

Control Valves ⋅ Cavitation in Control Valves
If oxygen is also released by the cavitation process.
50
SAMSON AG ⋅ V74/ Dob/Zin
. the oxidizing effect of the fluid increases. The contamination of the medium caused by cavitation-induced material erosion should not be underestimated in closed circuits.

and r is the distance between the sound source and the point of observation. t ) =
 ρF d 2 r  ⋅ 2 ⋅ V (t ) ⋅  t −  4πr dt cF  
(31)
In the equation. The spectral distribution of the energy E(f) of the noise created by a bubble is obtained from the squared Fourier transform of the pressure over time in the far field
 +∞  E (f ) =  ∫ p (t ) ⋅ e −2 πift dt   −∞ 
 r  while neglecting the retarded time  t −   cf 
2
(32)
SAMSON AG ⋅ 03/11
ρ ⋅ π 4 E (f ) =    ⋅f ⋅  r 
2
(∫
+∞
−∞
V (t ) ⋅ e −2 πift dt
)
2
(33)
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. V(t) is the rate of change in bubble volume which can be determined approximately using the Rayleigh-Plesset differential equation. The theoretical approaches to explain how noise develops are based on individual bubbles which implode concentrically in an infinitely expanded fluid and without affecting each other.Part 3 ⋅ L351 EN
Cavitation noise
The pressure peaks induced by the bubble implosions cause material erosion as well as a loud typical noise. The cavitation bubble can then be regarded as an isotropic radiator of zero order (monopole source) which creates in the far field the sound pressure
Cavitation bubbles act as an isotropic radiator of zero order
p (r .

As a consequence of the collapsing time being directly proportional to the bubble radius. the spectral energy decreases at f –2/5. it is possible to derive the total spectral sound energy Etotal(f) from the spectral energy of individual events E(f) and a mean number of bubble collapses per unit of time n.
Etotal (f ) = n ⋅ E (f )
(34)
Observations of individual bubbles only apply to incipient cavitation
These theoretical considerations made for individual bubbles apply approximately when cavitation just starts. or is proportional to the square of the differential pressure. C is the total volume of all bubbles per volume unit.
52
SAMSON AG ⋅ V74/ Dob/Zin
. As the cavitation progresses. Lyamshev [22] discovered on this basis that the sound intensity of a cavitation zone equals the flow velocity raised to the power of four. whose frequency follows a Poisson distribution.Control Valves ⋅ Cavitation in Control Valves
The spectral energy rises at first with the frequency f raised to the power of four and reaches a maximum at a frequency whose the inverse ratio is approximately the same as the collapsing time of the cavitation bubble. the maximum noise is shifted to lower frequencies as the bubble radius increases. and the contribution of each bubble to the entire noise depends on the history of the neighboring bubble. distinctive cavitation zones form in which the effects of each individual bubble overlap each other. At frequencies above the maximum. More extensive investigations therefore observe the cavitating fluid as a quasi-continuum of density
ρ=
ρF (1 + C )
(35)
In the equation. If the bubble collapses are observed as random events not connected with one another.

ρF ≈1000 kg/m3). The influence of the density and the differential pressure important for the course of the bubble implosion
The equations according to VDMA are based on empirical data
p 2 − pV =
1− xF ⋅ ∆p xF
(36)
are also only reproduced inexactly in the empirical VDMA equations which are based mainly on measurements with cold water (pV ≈ 0 bar. neither the influence of surface tension nor viscosity nor the influence of the gas contents are taken into account. The VDMA calculation methods [23] introduced in 1979 and improved in 1989 are.
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53
. but instead describe the course of internal sound power level on the basis of measurements related to xF. ρF and ∆p.Part 3 ⋅ L351 EN
The theoretical approaches were extended and refined by various authors. however. Despite this. In this case. Therefore. z. well proven and recognized worldwide. no one has yet successfully described the exceptionally complex correlations in the cavitation zone downstream of a control valve for various media and pressure ratios. The main features of the new standard are shown in [25]. the currently valid equations as per the German directive VDMA 24422 to predict noise emission of cavitating flows of control valves are not based on hydrodynamic and thermodynamic models. Note: The IEC 60534-8-4 international standard is currently being revised.

the acoustic power emitted in the octave band range of 0. Fig. load and xFZ value of the valve. 28 shows the course of the standardized sound power for standard valves with parabolic plugs (∆LF = 0) in relation to the operating pressure ratio with xFZ as a parameter.5 to 8 kHz in the pipeline is obtained according to
LWi = 134. The second row serves to calculate the sound power component which arises when the flow noise caused by the turbulence overlaps the cavitation noise resulting from the temporal static bubble collapse. on the compressibility of the fluid that increases with xF (see section on avoiding cavitation) and on the reduction of the driving force p2 – pV when the bubble implodes (equation 10). the noise emission rises steeply. When xF > xFZ applies.8
x FZ 0 . on the one hand. Cavitation behavior deviating from the calculation can be taken into consideration by the valve manufacturer by specifying a valve-specific correction coefficient ∆LF = f (xF.
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application in the octave band range 0.4 + 10 ⋅ lg Lwi − 120 ⋅
W ⋅ ∆p ⋅ ηF
ρF ⋅ FL 2
0. The drop in sound power level at high pressure ratios is based. reaches a maximum and drops back to the sound power level caused by the turbulence when xF = 1. The spectral distribution of the internal sound power level depends on the design.Control Valves ⋅ Cavitation in Control Valves
When cavitation (xF >> xFZ) occurs. y). According to VDMA 24422.5 to 8 kHz irrelevant of the operating
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. pressure ratios. + ∆LF 1 − x FZ (37)
with p ≤ FL 2 (p 1 − FF ⋅ p V )
The first row of the equation (37) represents the sound power level which is caused by the turbulent flow noise.0625 ⋅ (1 − x F ) x F x FZ
⋅ lg
1001 − x F . the spectral distribution can be reproduced for practical condition using a noise spectrum that drops 3 dB per octave.

there is the theory termed “hot spots” that assumes the gas compression linked with the bubble implosion leads to such extremely high temperatures that the gas starts to glow. The light appearances arise when the dipoles discharge on bubble implosion. In the cavitation zone. small flashes of light can be seen under certain conditions that have a spectral distribution from near infrared right up to the ultraviolet range. but is very suitable to demonstrate again the extreme. Others believe that the flashes of light occur when the free ions recombine which arise on bubble implosion due to the mechanical load of the molecules.
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SAMSON AG ⋅ V74/ Dob/Zin
.Control Valves ⋅ Cavitation in Control Valves
Cavitation luminescence
To conclude with. Finally. another cavitation phenomenon should be introduced which is insignificant in conjunction with control valves. Several authors trace the light emission back to the photochemical recombination of gas molecules that have thermally dissociated when the bubbles imploded. mostly uninvestigated conditions that exist when bubbles implode: cavitation luminescence. A further theory postulates aspherical bubbles which are surrounded by a layer of dipoles.